Index Notation, taking derivative

In summary, the derivative of (Aδij),j is (dp/dx_{1})A\delta_{i1}+ (dp/dx_{2})A\delta_{i2}+ (dp/dx_{3})A\delta_{i3}
  • #1
hellomrrobot
10
0
Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not sure if this would change anything).
 
Physics news on Phys.org
  • #2
You really should be using sub and super scripts because it becomes difficult to imagine what you intend, I am going to assume that you are meaning ##(A\delta_{ij})_{,j} = \partial_j A\delta_{ij}##.

The wonderful thing about index notation is that you can treat each term as if it was just a number and in the end you sum over repeated indices. So what you need to think about is what is the partial derivative ##\partial_k (A \delta_{ij})##. Once you have done that you can let ##k = j## and perform the sum.
 
  • #3
Orodruin said:
You really should be using sub and super scripts because it becomes difficult to imagine what you intend, I am going to assume that you are meaning ##(A\delta_{ij})_{,j} = \partial_j A\delta_{ij}##.

The wonderful thing about index notation is that you can treat each term as if it was just a number and in the end you sum over repeated indices. So what you need to think about is what is the partial derivative ##\partial_k (A \delta_{ij})##. Once you have done that you can let ##k = j## and perform the sum.

How would the partial derivative look like? This is what I got ##(dp/dx_{j}) \delta_{ij} + p (d \delta_{ij}/dx_{j})##
 
  • #4
The second term is zero, since ##\delta_{ij}## is a constant.
 
  • #5
Avodyne said:
The second term is zero, since ##\delta_{ij}## is a constant.

Okay so after I do the summation, I would get ##(dp/dx_{1})A\delta_{i1} + (dp/dx_{2})A\delta_{i2} + (dp/dx_{3})A\delta_{i3}##

Then if I set i=1, I would get just ##dp/dx_{1}## ?
 
  • #6
In general you can use the property ##\delta_{ij}A_j = A_i## as long as ##A_i## is any indexed expression.
 
  • #7
OP, I'm not sure what you're doing with the ##p## and the ##A##, so let's go back to basics:

Suppose ##A## is a scalar and ##T_{ij}## is a two-index tensor. Consider ##(AT_{ij})_{,k}##. (Later we will set ##T_{ij}=\delta_{ij}## and ##k=j##.) Then, using the product rule for derivatives, we have ##(AT_{ij})_{,k}=A_{,k}T_{ij}+A(T_{ij})_{,k}##. (As a side comment, the standard notation would be to write ##T_{ij,k}## instead of ##(T_{ij})_{,k}##.)

In the case of interest, we have ##T_{ij}=\delta_{ij}##. This is constant, and so its derivatives vanish: ##(\delta_{ij})_{,k}=0##. We are left with ##(A\delta_{ij})_{,k}=A_{,k}\delta_{ij}##.

Now let ##k=j##, with an implicit sum over the repeated ##j## index. Then we have ##(A\delta_{ij})_{,j}=A_{,j}\delta_{ij}##. We can perform the sum over ##j## using the general rule given by Orodruin: ##A_{,j}\delta_{ij}=A_{,i}##. So the final result is ##(A\delta_{ij})_{,j}=A_{,i}##.

Now you can set ##i## to a particular value (1, 2, or 3) if you like.
 

FAQ: Index Notation, taking derivative

1. What is index notation and how is it used?

Index notation is a mathematical notation system that is used to represent and manipulate algebraic expressions involving multiple variables. It uses subscripts to represent the different variables and their powers in an expression. This notation is commonly used in subjects such as calculus, physics, and engineering.

2. How do you take the derivative using index notation?

To take the derivative of an expression using index notation, you first identify the variable that you are differentiating with respect to and its corresponding subscript. Then, you multiply the coefficient of the variable by its subscript and decrease the subscript by 1. This process is repeated for each term in the expression.

3. What is the advantage of using index notation to take derivatives?

The advantage of using index notation to take derivatives is that it allows for a more compact and organized representation of expressions, especially when dealing with multiple variables. It also makes it easier to apply the rules of differentiation and simplifies the process of taking derivatives of complex expressions.

4. Can you use index notation to take derivatives of non-polynomial functions?

Yes, index notation can be used to take derivatives of non-polynomial functions such as exponential, logarithmic, and trigonometric functions. The same rules of differentiation apply, but the subscripts may be more complex depending on the specific function.

5. Are there any limitations to using index notation for taking derivatives?

One limitation of using index notation for taking derivatives is that it cannot be used for functions with discontinuities or undefined points. In these cases, other techniques such as limit definitions or the chain rule must be used to find the derivative. Additionally, index notation may become more complex and difficult to use for very large or complicated expressions.

Similar threads

Back
Top